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If the abscissae and the ordinates of tw...

If the abscissae and the ordinates of two point `Aa n dB` be the roots of `x^2+2ax−b^2 `a n d `x^2+2px−q^2=0` respectively, show that the equation of the circle described on `A B` as diameter is `x^2+y^2+2ax+2py−b^2−q^2=0`

Text Solution

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Given,`x^2+2ax−b^2=0`
Let roots be `x_1`​ & `x_2`​
`x_1​+x_2​=(−b)/a​=−2a`
`x_1​x_2​=c/a​=−b^2`
...
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If the abscissa and ordinates of two points P and Q are the roots of the equations x^(2)+2ax-b^(2)=0 and x^(2)+2px-q^(2)=0 respectively,then find the equation of the circle with PQ as diameter.

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Knowledge Check

  • If the abscissas and ordinates of two points P and Q are the roots of the equations x^(2)+2ax-b^(2)=0 and x^(2)+2px-q^(2)=0 respectively, then equation of the circle with PQ as diameter is

    A
    `x^(2)+y^(2)+2ax+2py-b^(2)-q^(2)=0`
    B
    `x^(2)+y^(2)-2ax-2py+b^(2)+q^(2)=0`
    C
    `x^(2)+y^(2)-2ax-2py-b^(2)-q^(2)=0`
    D
    `x^(2)+y^(2)+2ax+2py+b^(2)+q^(2)=0`

  • Abscissas and ordinates of points A and B are roots of equation x^(2) + 2ax - b^(2) = 0 " and " y^(2) + 2py - q^(2) = 0 respectively. Equation of circle with AB as a diameter is

    A
    `x^(2) + y^(2) + 2ax + 2py + b^(2) + q^(2) = 0 `
    B
    `x^(2) + y^(2) - 2ax - 2py - b^(2) - q^(2) = 0 `
    C
    `x^(2) + y^(2) - 2ax + 2py - b^(2) -q^(2) = 0`
    D
    `x^(2) + y^(2) - 2ax - 2py + b^(2) + q^(2) = 0 `

  • If abscissas and ordinates of points A , B are roots of equation x^(2) - 3x + 2 = 0 " and " y^(2) - 7y + 12 = 0 , then equation of circle with AB as a diameter is

    A
    `x^(2) + y^(2)- 3x + 7y + 14 = 0`
    B
    `x^(2) + y^(2) + 3x - 7y + 14 = 0 `
    C
    `x^(2) + y^(2) + 3x + 7y + 14 = 0 `
    D
    `x^(2) + y^(2) - 3x - 7y + 14 = 0 `

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    The abscissa of two points A and B are the roots of the equation x^(2)+2ax-b^(2)=0 and ordinates are the roots of equation x^(2)+2ax-b^(2)=0 . Find the equation of the circle whose diameter is AB.

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